On majorization and Schur products

Abstract

Suppose A, D 1,…, D m are n × n matrices where A is self-adjoint, and let X = Σ m k = 1D kAD ∗ k . It is shown that if ΣD kD ∗ k = ΣD ∗ kD k = I , then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D 1,…, D m , a result is obtained in terms of weak majorization. If each D k is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if b ii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved.